Mean concentration instantaneous source

III. Mean Concentration from an Instantaneous Source in Stationary,... 

Let us consider, as we did in the previous section, an instantaneous point source of strength Q at the origin in an infinite fluid with a velocity u in the x direction. The mean concentration in the case of constant K x, Kyy, and is governed by... 

We note the similarity of Eqs. (3.27) and (3.15). In particular, if we define o- = 2X t, o-y = 2Kyyt, and = 2K t, the two expressions are identical when y = w = 0. Thus, we see that the mean concentration from an instantaneous point source in an infinite fluid with stationary, homogeneous turbulence has a Gaussian form, with the variances of the concentration distribution related to the variances of the wind velocity fluctuations or to constant eddy difihisivities. 

Solutions were obtained in Section III for the mean concentration resulting from an instantaneous release of a quantity Q of material at the origin in an infinite fluid with stationary, homogeneous turbulence and a mean velocity in the x direction. We now wish to consider the case of a continuously emitting source under the same conditions. The source strength is specified as q (g sec )-... 

This example can readily be generalized to three dimensions. If we continue to assume that there is a mean flow only in the x direction, then the expression for the mean concentration resulting from an instantaneous point source of unit strength at the origin is... 

Up to this point in this chapter we have developed the common theories of turbulent diffusion in a purely formal manner. We have done this so that the relationship of the approximate models for turbulent diffusion, such as the K theory and the Gaussian formulas, to the basic underlying theory is clearly evident. When such relationships are clear, the limitations inherent in each model can be appreciated. We have in a few cases applied the models obtained to the prediction of the mean concentration resulting from an instantaneous or continuous source in idealized stationary, homogeneous turbulence. In Section 18.7.1 we explore further the physical processes responsible for the dispersion of a puff or plume of material. Section 18.7.2 can be omitted on a first reading of this chapter that section goes more deeply into the statistical properties of atmospheric dispersion, such as the variances a (r), which are needed in the actual use of the Gaussian dispersion formulas. 

Fig. 8.1 An image of an odor plume taken using planar, laser-induced fluorescence. This image reveals the instantaneous scalar structure of the plume. The image was captured from the outer layer of the momentum boundary layer of the plume. It is a horizontal image spanning a lateral and streamwise range it reveals the spatial patterns at a given vertical location. The color scale indicates the concentration of the odor in the plume concentrations are normalized by the source concentration Co and color coded as shown in the legend. From Grimaldi et al.. Journal of Turbulence, 2002, The relationship between mean and instantaneous structure in turbulent passive scalar plumes, vol. 3, pp. 1-24. Reproduced with the permission of the authors and Taylor and Francis Ltd. (www.tandf.co.uk/ioumals).

Thus far the only difference between the UV and El detectors is that the observed signal for the former is a measure of transmitted intensity I, while that for the latter is a measure of the absorbed intensity — but it turns out that is of little consequence for the present purpose. Thus, for example, a fluorescence detector records a signal that is a measure of the absorbed intensity of the exciting radiation like the El case, but it is also a concentration dependent detector like the simple UV absorption detector. The difference between the two types arises rather in the relationship between the analyte concentration delivered by the mobile phase (c ) and that within the absorption cell or El source in the former case c = c (see equation [4.2]), but the situation is very different in the El case where the mass spectrometer vacuum pumps continuously remove the analyte from the El source. In fact c ei represents an instantaneous steady state value, a compromise between the flow rate of A into the source and the pumping rate out of the source here instantaneous means simply that the establishment of the steady state value c gi occurs on a timescale appreciably shorter than that of the chromatographic peak. Then at this steady state ... 

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